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Vibration Transmission of a Cylindrical Shell with an Interior Rectangular Plate with the Receptance Method  [PDF]
Zhimei Zhao,Meiping Sheng,Yang Yang
Advances in Acoustics and Vibration , 2012, DOI: 10.1155/2012/581769
Abstract: Vibration transmission characteristics of a cylindrical shell with a rectangular plate are discussed in this paper by the substructure receptance method. The system is divided into two substructures: the plate and the shell. After finding the theoretical receptance function of each substructure, the coupling equation of the combined system is solved by considering the continuity conditions at the joint between the plate and the shell. The numerical results are compared with the experimental ones to show the validity of the formulation. After that, effects of the plate's parameters on vibration transmission characteristics are discussed. The parameter study shows that it is important to avoid impedance matching for the plate and the shell in the design of suitable structures with low vibration and noise radiation characteristics. 1. Introduction Plates and shells are widely used as basic components in many engineering structures, such as aircrafts [1] and underwater structures [2]. Until now, their fundamental physical mechanisms and phenomenons, such as free vibration [3], wave propagation [4, 5], and power flow [6], have been discussed intensely. Plates and shells are often utilized by coupling together with welds, bolts, or dampers in practical applications. Dynamic behaviors of these coupled structures become relatively complicated due to vibration energy transmission between the interior plate and the shell. Less literature is available on these topics. However, as vibration transferred from the plate to the shell would result in the shell’s outward noise radiation; it is of great importance for engineers to understand these combined structures’ vibration, especially the effects of parameters on the combinations’ dynamic behaviors, which is helpful for designing suitable structures with low vibration and noise radiation. Forced vibration of one variant of these combined structures, a partitioned cylindrical shell with a longitudinal, interior rectangular plate, has been investigated in this paper to simulate an aircraft fuselage with an interior floor. Vibroacoustic characteristics of a cylindrical shell with an interior rectangular plate have been studied in several papers, which mainly focused on free vibration analysis. The first analytical model was developed by Peterson and Boyd [7] with applications of Rayleigh-Ritz technique. Effects of structural parameters, including rigid joint, hinged joint, thickness and position of the floor on natural frequencies and mode shapes were analyzed. Irie et al. [8] studied free vibration of noncircular
Finite Element Formulation of Forced Vibration Problem of a Prestretched Plate Resting on a Rigid Foundation  [PDF]
M. Er z,A. Yildiz
Journal of Applied Mathematics , 2007, DOI: 10.1155/2007/56360
Abstract: The three-dimensional linearized theory of elastodynamics mathematical formulation of the forced vibration of a prestretched plate resting on a rigid half-plane is given. The variational formulation of corresponding boundary-value problem is constructed. The first variational of the functional in the variational statement is equated to zero. In the framework of the virtual work principle, it is proved that appropriate equations and boundary conditions are derived. Using these conditions, finite element formulation of the prestretched plate is done. The numerical results obtained coincide with the ones given by Ufly and in 1963 for the static loading case.
THREE-DIMENSIONAL EXACT SOLUTION FOR FREE AND FORCED VIBRATIONS OF MULTILAYERED PLATES
层合板自由振动和强迫振动的三维精确解

YANG Yazheng,LIU Hua,
杨亚政
,刘华

力学与实践 , 2008,
Abstract: Based on the three-dimensional theory of elasticity and the state space approach, this paper studies the free vibration and forced vibration of an isotropic multilayered rectangular plate with four edges simply supported. Firstly, on the basis of the theory of elasticity, the governing equations of a multilayered plate are established and the free vibration response of the laminated panel is obtained by the state space approach. Then the dynamic response of the multilayered plate subjected to an impact loading is obtained from Lagrangian dynamic formulations.
运动薄膜的速度对非线性强迫振动的影响研究
The Effect of Moving Membrane Velocity on Nonlinear Forced Vibration
 [PDF]

邵明月,武吉梅,王砚,应戍狄,赵凡
- , 2018,
Abstract: 研究了运动薄膜的速度对非线性强迫振动的影响。基于Von Karman薄板理论推导出轴向运动薄膜大挠度振动方程,应用Galerkin方法对振动偏微分方程组进行离散,得到系统的状态方程,采用4阶Runge-Kutta法对系统状态方程进行数值求解,利用分岔图分析了薄膜非线性振动特性与速度的关系,得到了薄膜产生混沌的区间和稳定工作区间。通过时程图、相图、Poincare截面图和功率谱分析系统的周期运动和混沌运动。
The effect of the velocity of moving membrane on the nonlinear forced vibration is investigated in this paper. Large deflection vibration equation of an axially moving membrane is deduced by using the Von Karman thin plate theory. Galerkin method is applied to discretize vibration differential equations of the membrane, and then the state equation of the system is obtained. The state equation of the system is numerically solved by the fourth order Runge-Kutta method, and the relationship between the nonlinear vibration characteristics and the velocity of the membrane is analyzed by using the bifurcation diagram. The chaotic region and stable working region of the moving membrane are obtained. The periodic motion and chaos motion of the system are analyzed by time histories, phase-plane portraits, Poincare maps and power spectrum
Non-linear forced vibration analysis of rectangular plates including the coupling between transverse and in-plane displacements.  [cached]
Merrimi E.,El Bikri K.,Benamar R.
MATEC Web of Conferences , 2012, DOI: 10.1051/matecconf/20120110014
Abstract: The objective of this work is to study the geometrically non-linear steady state periodic forced response of fully clamped rectangular plates (FCRP) with immovable in-plane conditions, taking into account the effect of the in-plane displacements. A complete formulation has been proposed first, reducing the equations of motion to a system of coupled non-linear algebraic equations, which are decoupled once the in-plane inertia is omitted. An averaging technique has then been developed, in order to simplify the first method and to develop an engineering complete theory. The forced response is given in the case of a concentrated harmonic excitation force with various intensities. The numerical results obtained here with the two formulations, using an explicit analytical solution, were compared with those obtained previously using a formulation in which the in-plane displacements have been neglected, showing an “over-stiff” effect.
Parametric Study of Nonlinear Beam Vibration Resting on Linear Elastic Foundation
Journal of Mechanical Engineering and Automation , 2012, DOI: 10.5923/j.jmea.20120206.02
Abstract: Nonlinear beam resting on linear elastic foundation and subjected to harmonic excitation is investigated. The beam is simply supported at both ends. Both linear and nonlinear analyses are carried out. Hamilton’s principle is utilized in deriving the governing equations. Well known forced duffing oscillator equation is obtained. The equation is analyzed numerically using Runk-Kutta technique. Three main parameters are investigated: the damping coefficient, the natural frequency, and the coefficient of the nonlinearity. Stability regions for first mode analyses are unveiled. Comparison between the linear and the nonlinear model is presented. It is shown that first mode shape the natural frequency could be approximated as square root of the sum of squares of both natural frequency of the beam and the foundation. The stretching potential energy is proved to be responsible for generating the cubic nonlinearity in the system.
Forced Vibration Numerical Analysis of Rectangular Elastic Orthotropic Damped Inclined Mindlin Plate Using Finite Difference Algorithm  [PDF]
Michael C. Agarana, Amenawon I. Ehigbochie
Applied Mathematics (AM) , 2018, DOI: 10.4236/am.2018.96043
Abstract:
Plates vibrate when load moves on them. In this paper, the dynamic response of Mindlin plate analytical model was converted to its numerical form using finite difference algorithm. The numerical model was analysed to ascertain the critical parameters contributing to the deflection of Mindlin plate under a moving load. The examination was more reasonable as in the likelihood of the plate laying on a Pasternak foundation was put into thought. Likewise the impact of damping was not dismissed. The plate considered in this paper was an inclined Mindlin plate, where the impacts of shear deformation and rotatory inertia were considered. The numerical equations were solved with the help of a developed computer program and Matlab. The results were consistent with what we have in the literature. The effects of the Pasternak foundation, damping, angle of inclination, and the moving load to the dynamic response of the elastic plate were exceptionally self-evident.
ON THE PHENOMENA OF CHAOS IN A FORCED VIBRATION SYSTEM WITH A SQUARE NONLINEAR TERM
关于含二次非线性项受迫振动系统的混沌现象

CHEN LI-QUN,
陈立群

物理学报 , 1989,
Abstract: In this paper, the threshold of chaos in a forced vibration system with a square nonlinear term is given by using the Melnikov method. The effects of parameters on the threshold are analyzed. The known results obtained respectively by approximate 2-D iterated map and by numerically integrating the differential equation are compared with the results here.
Nonlinear Forced Vibration of a Viscoelastic Buckled Beam with 2 : 1 Internal Resonance  [PDF]
Liu-Yang Xiong,Guo-Ce Zhang,Hu Ding,Li-Qun Chen
Mathematical Problems in Engineering , 2014, DOI: 10.1155/2014/906324
Abstract: Nonlinear dynamics of a viscoelastic buckled beam subjected to primary resonance in the presence of internal resonance is investigated for the first time. For appropriate choice of system parameters, the natural frequency of the second mode is approximately twice that of the first providing the condition for 2?:?1 internal resonance. The ordinary differential equations of the two mode shapes are established using the Galerkin method. The problem is replaced by two coupled second-order differential equations with quadratic and cubic nonlinearities. The multiple scales method is applied to derive the modulation-phase equations. Steady-state solutions of the system as well as their stability are examined. The frequency-amplitude curves exhibit the steady-state response in the directly excited and indirectly excited modes due to modal interaction. The double-jump, the saturation phenomenon, and the nonperiodic region phenomena are observed illustrating the influence of internal resonance. The validity range of the analytical approximations is assessed by comparing the analytical approximate results with a numerical solution by the Runge-Kutta method. The unstable regions in the internal resonance are explored via numerical simulations. 1. Introduction The main goal of this paper is to present an exhaustive investigation for the forced vibration of a buckled beam with quadratic and cubic nonlinearities in the equations of motion. Buckled beams play an important role in the design of machines and structures. Buckled beams have received a great deal of attention from various scholars also due to their complex nonlinear behaviors [1]. For example, the nonlinear modal interaction [2] and the internal resonance [3] are arising out of commensurable relationships of frequencies for specific values of the system parameters. Furthermore, in the presence of external excitation, the internal resonance can have possible influence on the buckled beam behavior, which needs to be studied [4]. Lacarbonara et al. [5] studied the frequency-response curves of a primary resonance of a buckled beam. The authors found that the response curves obtained using a single-mode approximation are in disagreement with those obtained by the experiment. Emam and Nayfeh [6] focused on the dynamics of a buckled beam subject to a primary-resonance excitation via the Galerkin truncation. They revealed that using a single-mode truncation leads to errors in the static and dynamic problems. Nayfeh and Balachandran pointed out that those systems are characterized by quadratic nonlinearities which
Thermal Effect on Vibration of Tapered Rectangular Plate  [PDF]
Anupam Khanna,Ashish Singhal
Journal of Structures , 2014, DOI: 10.1155/2014/642926
Abstract: A mathematical model is constructed to help the engineers in designing various mechanical structures mostly used in satellite and aeronautical engineering. In the present model, vibration of rectangular plate with nonuniform thickness is discussed. Temperature variations are considered biparabolic, that is, parabolic in x-direction and parabolic in y-direction. The fourth-order differential equation of the motion is solved by Rayleigh Ritz method for three different boundary conditions around the boundary of plate. Numerical values of frequencies for the first two modes of vibration are presented in tabular form for different values of thermal gradient, taper constants, and aspect ratio. 1. Introduction The structures are designed to support the high speed engines and turbines subjected to the vibration. Due to faulty design, there is unbalance in the engines which causes excessive and unpleasant stresses because of vibration. In the field of science and technology, it is preferred to design large machines and structures for smooth operations with controlled vibration. Recent studies in the field of vibrational behavior create a huge interest for scientists and engineers in the designs and constructions of complex systems or structures such as ships, submarines, aircrafts, launch vehicles, missiles, and satellites. Tapered plates or plates of nonuniform thickness are commonly used in many engineering applications such as nuclear engineering, aeronautical engineering, and chemical plants, under the influence of elevated temperature to control high vibration. A collection of research papers on vibration of plates with different shapes and boundary conditions is given by Leissa [1] in his monograph. Leissa [2] discussed different models on free vibration of rectangular plates. Jain and Soni [3] analyzed the free vibrations of rectangular plates with parabolically varying thickness. Tomar and Gupta [4] studied the effect of thermal gradient on the vibration of a rectangular plate with bidirectional variation in thickness. Leissa [5] investigated the effect of thermal gradient on the vibration of parallelogram plate with bidirectional thickness variation in both directions. Singh and Chakraverty [6] studied the transverse vibration of circular and elliptical plates with variable thickness. Sharma and Chand [7] analyzed the vibrations in transversely isotropic plates due to suddenly punched hole. Leissa [8] discussed the historical bases of the Rayleigh and Ritz methods for the vibrations of plates. Chakraverty et al. [9] studied the flexural vibrations of
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